A Floor That Gives Way

solved NewtonsSecondLaw

Q. A wooden block of 2 kg $m_{b} = 2$ kg rests on a soft horizontal floor. When an iron cylinder of 25 kg $m_{c} = 25$ kg is placed on top of the block $m = m_{b} + m_{c}$ , the floor yields steadily and the block and the cylinder together go down $a = - | a |$ with an acceleration of 0.1 m/s² $| a | = 0.1$ m/s². Taking g as $W = - | W |$ 10 m/s² $g = 10$ m/s², how hard does the floor push up on them once it yields, in newtons?

Inputs

block mass

m_{b}

2 kg

cylinder mass

m_{c}

25 kg

acceleration

| a |

0.1 m/s²

gravity

g

10 m/s²

Outputs

push

| F_{f} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m_{b} = 2$ kg
$m_{c} = 25$ kg
$| a | = 0.1$ m/s²
$g = 10$ m/s²
$W = - | W |$ from “Taking g as”
$m = m_{b} + m_{c}$ from “placed on top of the block”
$a = - | a |$ from “go down”

Find $| F_{f} |$ .

Working

placed on top of the block $m = m_{b} + m_{c}$
$m = 2 + 25$
$m = 27$ kg
go down $a = - | a |$
$a = - 0.1$
$a = −0.1$ m/s²
second law(sinking, y) $F_{net} = m \cdot a$
$F_{net} = 27 \cdot (−0.1)$
$F_{net} = −2.7$ N
weight of a body(pair) $| W | = m \cdot g$
$| W | = 27 \cdot 10$
$| W | = 270$ N
Taking g as $W = - | W |$
$W = - 270$
$W = −270$ N
net force is the resultant(pair, sinking, y) $F_{net} = F_{f} + W$
$(−2.7) = F_{f} + (−270)$
$F_{f} = 267.3$ N
magnitude from components(force on pair by floor during sinking) $| F_{f} | = \sqrt{{F_{f}}^{2}}$
$| F_{f} | = \sqrt{{267.3}^{2}}$
$| F_{f} | = 267.3$ N
$| F_{f} | \approx 267.3$ Nto 4 significant figures

Conclusion

The floor holds up less than the full weight, by just what it takes to let the two sink together; by the third law they press down on the floor with the same force.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_floor_that_gives_way do4  kind :numerical5  describes "A wooden block of %{block_mass} rests on a soft horizontal floor. When an iron cylinder of " \6            "%{cylinder_mass} is placed on top of the block, the floor yields steadily and the block and the " \7            "cylinder together go down with an acceleration of %{acceleration}. Taking g as %{gravity}, " \8            "%{push}, in newtons?"9  given block_mass: "2 kg"10  given cylinder_mass: "25 kg"11  given acceleration: "0.1 m/s2"12  given gravity: "10 m/s2"13  asks push: "how hard does the floor push up on them once it yields"14end

solution.rb

1require_relative "../questions/a_floor_that_gives_way"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_floor_that_gives_way, using: NewtonsSecondLaw do5  objects :pair, :block, :cylinder, :floor6  axes :y7  pair.goes(:sinking)8  deduce_from "Taking g as", pair.weight.direction => -y910  take block_mass: block.mass11  take cylinder_mass: cylinder.mass12  take acceleration: pair.acceleration.magnitude13  take gravity: g1415  deduce_from "placed on top of the block", pair.mass => block.mass + cylinder.mass16  deduce_from "go down", pair.acceleration.direction => -y1718  push = pair.force(by: floor).magnitude19  answer push:, in: "N", figures: 42021  conclusion "The floor holds up less than the full weight, by just what it takes to let the two sink " \22             "together; by the third law they press down on the floor with the same force."23end

playground.rb

1require_relative "../solutions/a_floor_that_gives_way"23play_with :a_floor_that_gives_way do4  title "A Floor That Gives Way"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$F$	N	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	$v_{av}$	m/s	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
gravity	$g$	m/s²	—	UniformAcceleration
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
displacement of pair during sinking along y	$Δ x$	m	—
position of pair at end of sinking along y	$x_{1}$	m	—
position of pair at start of sinking along y	$x_{0}$	m	—
average velocity of pair during sinking along y	$v_{av}$	m/s	—
time taken of sinking	$Δ t$	s	0 … 1e+05
average speed of pair during sinking	$c_{av}$	m/s	—
path length of pair during sinking	$d$	m	—
change of velocity of pair during sinking along y	$Δ v$	m/s	—
velocity of pair at end of sinking from ground along y	$v_{1}$	m/s	—
velocity of pair at start of sinking from ground along y	$v_{0}$	m/s	—
average acceleration of pair during sinking along y	$a_{av}$	m/s²	—
acceleration of pair during sinking along y	$a$	m/s²	—
net force on pair during sinking along y	$F_{net}$	N	—
mass of pair	$m$	kg	0 … ∞
mass of block	$m_{b}$	kg	0 … ∞
mass of cylinder	$m_{c}$	kg	0 … ∞
mass of floor	$m_{f}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on pair	$\| W \|$	N	—
weight on block	$\| W_{b} \|$	N	—
weight on cylinder	$\| W_{c} \|$	N	—
weight on floor	$\| W_{f} \|$	N	—
force on pair by floor during sinking	$\| F_{f} \|$	N	—
weight on pair along y	$W$	N	—
force on pair by floor during sinking along y	$F_{f}$	N	—
displacement of pair during sinking	$\| Δ x \|$	m	—
position of pair at end of sinking	$\| x_{1} \|$	m	—
position of pair at start of sinking	$\| x_{0} \|$	m	—
average velocity of pair during sinking	$\| v_{av} \|$	m/s	—
change of velocity of pair during sinking	$\| Δ v \|$	m/s	—
velocity of pair at end of sinking from ground	$\| v_{1} \|$	m/s	—
velocity of pair at start of sinking from ground	$\| v_{0} \|$	m/s	—
average acceleration of pair during sinking	$\| a_{av} \|$	m/s²	—
acceleration of pair during sinking	$\| a \|$	m/s²	—
net force on pair during sinking	$\| F_{net} \|$	N	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

v_{0} \equiv v_{i} (start of motion)

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

a_{i} (part 1) = a_{i} (whole)

steady through its parts

m_{b} \neq 0

has mass?

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y

Δ x = x_{1} - x_{0}

change of position(sinking, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(sinking, y)

c_{av} = \frac{d}{Δ t}

path a time(sinking, y)

Δ v = v_{1} - v_{0}

velocity gained(sinking, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(sinking, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(sinking, y)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(sinking, y)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(sinking, y)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(sinking, y)

F_{net} = m \cdot a

second law(sinking, y)

| W | = m \cdot g

weight of a body(pair)

F_{net} = F_{f} + W

net force is the resultant(pair, sinking, y)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of pair during sinking)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of pair at end of sinking)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of pair at start of sinking)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of pair during sinking)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of pair during sinking)

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of pair at end of sinking from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of pair at start of sinking from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of pair during sinking)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of pair during sinking)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on pair during sinking)

| W | = \sqrt{W^{2}}

magnitude from components(weight on pair)

| F_{f} | = \sqrt{{F_{f}}^{2}}

magnitude from components(force on pair by floor during sinking)

W = - | W |

Taking g as

m = m_{b} + m_{c}

placed on top of the block

a = - | a |

go down

m \neq 0

has mass?(pair)

m_{b} \neq 0

has mass?(block)

m_{c} \neq 0

has mass?(cylinder)

m_{f} \neq 0

has mass?(floor)

F_{net} = 0

free of forces?(pair, sinking)